# Full text of "Treatise On Analysis Vol-Ii"

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21 PRODUCT OF MEASURES 219 (c) Show that there exists a measurable subset A of X with measure >0, such that for each XQ e A there exists a function g e JS?£(X, ju) (depending on x0) for which the series with general term (g \fn)fn(x0) does not converge. (Take A to be the set of points x e X for which the sequence (/„(*)) does not converge to 0, and use Problem 7(a) of Section 13.11. Then argue by contradiction, using (a) and (b).) 5. Let S be a closed subset of R. Given a sequence (cn)n^0 of real numbers, show that there exists a positive measure ju on R with support contained in S and such that I x" dp(x) = cn for all n S> 0 if and only if, for every polynomial P(X) = £ &X* such J * = o that P(x) ;> 0 for all *sS, we have £ f*c*^0. (As in Problem 2 of Section 13.3, fc = 0 show that there exists a positive measure /JL with support S which extends to a linear form u defined on «#"R(R) and on the space of polynomials on R, and such that w(P) = Z £kCk for each polynomial P(X) = £ & X*. Then prove that each power xn * = 0 fc=»0 is /x-integrable and that J xn d^x) = cn. For this purpose, remark that for each in- teger n ^ 0 and each £ > 0 there exists a number R > 0 such that |/n,R(*)| <^ ex2n+2, where /„, R is the function equal to 0 for \x\ < R and to x* for \x\ ;> R.) Particular cases: (1) S = R ("Hamburger's moment problem"): the condition is n that the quadratic forms £ c/+*6£* should be positive for ail n (observe that every J.fcsaO polynomial P(#) which is ^0 on R is the sum of two squares (P^x))2 + (P2Ct))2). (2) S = [0, 4- oo [ 0' Stieltjes* moment problem "): the condition is that the quad- n n ratic forms ]£ Cj+k£j£k and ^ cj+k+i^j^k should be positive for all n> 0 (remark J, *c = 0 J, *=»0 ~~ that every polynomial P(x) which is i> 0 on [0, + oo [ can be written in the form (Pi(x))2 4- (P2W)2 + *((P3(*))2 + (P^C*))2). 6. Let v be a positive measure on X and (/„) a sequence of functions belonging to , v) such that the sequence of measures (fn-v) converges to f-v (where , v)) for the topology ST^. Show that f(x) ^ lim sup/n(x) almost everywhere with respect to v. (Use (13.8.3).) 7. Let ju,, v be two bounded complex measures on X. Show that the following conditions are equivalent: (1) ^ and v are disjoint; (2) ||/u±v|| = \\p\\ + ||v||; ||v||). 21. PRODUCT OF MEASURES (13.21.1) Let X, Y be two locally compact spaces, 1 a measure on X and \i a measure on Y. Then there exists a unique measure v on the product space X x Y such that, for each pair of functions fe Jf C(X) andg G Jfc(Y), we have (13.21.11) 0. (Reduce to the case where g is bounded and hence